Download Canonicity and representable relation algebras

What am I going to do?
Canonicity and
representable relation algebras
Ian Hodkinson
Joint work with:
Rob Goldblatt
Robin Hirsch
Yde Venema
In the 1960s, Monk proved that the variety RRA of representable
relation algebras is canonical.
This is an instance of general results on canonical varieties due to
Fine, van Benthem, and Goldblatt (1975–89).
Recent work has shown that RRA only just manages to be canonical:
• it has no canonical axiomatisation,
• any first-order axiomatisation of it has infinitely many
non-canonical axioms.
This is connected to a recent counterexample to the converse of the
Fine–van Benthem–Goldblatt theorems.
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1. Relation algebras
These are algebras
Outline
,
A = (A, +, −, 0, 1, 1 , ˘, ;)
1. Relation algebras; representations
2. Duality: atom structures, complex algebras
satisfying certain equations laid down by Tarski in 1941.
(We will only need that (A, +, −, 0, 1) is a boolean algebra.)
3. Canonical extensions; canonicity of RRA
4. RRA has no canonical axiomatisation
RA denotes the class (variety) of relation algebras.
5. More
Relation algebras are examples of Boolean algebras with operators
(BAOs).
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Representations; RRA
Intended examples of relation algebras:
Re(U ) = (℘(U × U ), ∪, \, ∅, U × U, IdU , −−1 , | )
where
IdU
= {(x, x) : x ∈ U },
R−1
= {(y, x) : (x, y) ∈ R},
R|S
= {(x, y) : ∃z((x, z) ∈ R ∧ (z, y) ∈ S)}.
A representation of a relation algebra A is an embedding
Q
h : A → i∈I Re(Ui ) for some sets Ui (assumed pairwise disjoint).
h provides a ‘representation’ of each a ∈ A as a binary relation h(a)
S
on the base set i∈I Ui .
Some facts about RRA
Lyndon (1950): RRA ⊂ RA.
Tarski (1955): RRA is a variety — equationally axiomatised.
Monk (1964): RRA is not finitely axiomatisable.
Generally, RRA is rather wild.
Most questions about it seem to have the answer ‘no’.
A is said to be representable if it has a representation.
RRA denotes the class of representable relation algebras.
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2. Duality: atom structures and complex algebras
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3. Canonical extensions (Jónsson–Tarski, 1951)
Consider relation symbols R1, (unary), R˘ (binary), R; (ternary).
An atom structure is a relational structure of this type.
An ultrafilter of a relation algebra A is an uf. of its boolean reduct.
Given an atom structure S = (A, R1, , R˘, R; ), we can form its
,
complex algebra Cm S = (℘(A), ∪, \, ∅, A, 1 , ˘, ;), where
The set of ultrafilters of A forms an atom structure A+ .
Operations: e.g., A+ |= R; (α, β, γ) iff a ∈ α, b ∈ β ⇒ a ; b ∈ γ.
,
1
= {x ∈ A : S |= R1, (x)},
X̆
= {y ∈ A : ∃x ∈ X(S |= R˘(x, y))}.
X ;Y
The canonical extension of A is Aσ = Cm(A+ ).
(A embeds canonically into Aσ , so this is an extension of A.)
= {z ∈ A : ∃x ∈ X ∃y ∈ Y (S |= R; (x, y, z))}.
Under certain conditions on S, Cm S will be a relation algebra.
Any atomic relation algebra A has an atom structure, At A.
If A is complete atomic, Cm At A ∼
= A.
Again, this can be done for BAOs.
For BAOs (A, +, −, 0, 1, ♦) (♦ a unary function), it’s close to the
famous canonical model construction in modal logic (Lemmon, 1966)
— useful in proving completeness of axioms over a class of frames.
This can all be done for BAOs too.
E.g., we can form Cm S from an atom structure (‘frame’) S.
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Why is canonicity of RRA interesting?
Canonicity of RRA
• One of the few positive properties of RRA.
σ
A class K of relation algebras/BAOs is canonical if A ∈ K ⇒ A ∈ K.
Theorem 1 (Monk, 1960s) RRA is canonical.
Monk’s proof was reported in McKenzie’s Ph.D. thesis (1966).
We can now prove it using general results of Fine, van Benthem, &
Goldblatt (1975–89) on canonicity of modal logics/varieties of BAOs.
Fact 2 (Goldblatt) If K is an elementary class of atom structures,
then the variety HSP{Cm S : S ∈ K} generated by K is canonical.
Proof. [of theorem 1] By definition, RRA is generated by
K = {At Re(U ) : U a set}. But K is elementary (exercise).
• Easier to work with A+ than A.
• If A ∈ RRA then Aσ is ‘completely representable’ — it has a
representation respecting all infs and sups in Aσ .
So canonical extensions of RRAs are rather well behaved.
• But, unlike in modal logic, knowing RRA is canonical doesn’t
seem to help to axiomatise it.
• It turns out that RRA is only barely canonical.
The connection between A being representable and Aσ being
representable is rather loose.
The rest of the talk is mainly about this.
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Sahlqvist axioms
4. Canonical axioms for RRA?
An axiom τ of the signature of relation algebras is canonical if
A |= τ ⇒ Aσ |= τ .
Any canonically axiomatised variety is canonical.
But the converse may not hold. Canonicity might come about ‘in the
limit’ — an ‘emergent property’ of a set of non-canonical axioms.
Question 3 (Venema, ∼1995) Does RRA have a canonical
axiomatisation – each individual axiom in it is canonical?
Bit different from usual questions about unpleasantness of
axiomatisations of RRA.
It is undecidable whether an equation is canonical.
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After 1997, Venema conjectured ‘no’, because of:
Fact 4 (Venema, 1997) RRA has no Sahlqvist axiomatisation.
Sahlqvist equations are defined syntactically. They are paradigm
examples of canonical axioms. E,g., all positive equations are
Sahlqvist.
Venema knew Sahlqvist equations are preserved under (Monk)
completions of relation algebras. So fact 4 follows from
Fact 5 (IH, 1997) RRA is not closed under completions.
(This answered an implicit question of Monk (1970).)
But arbitrary canonical equations might not be preserved under
completions. So how can we confirm Venema’s conjecture?
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Key idea
Take a fixed axiomatisation Γ of RRA.
Measure representability of a relation algebra by how much of Γ it
satisfies.
Canonicity of RRA: if A is fully representable (A ∈ RRA), so is Aσ .
By first-order compactness, if there is a canonical axiomatisation of
RRA, then Aσ can be made as representable as we like, by making
A sufficiently representable.
So we construct relation algebras A of unbounded representability,
whilst the representability of Aσ is bounded.
This shows RRA has no canonical axiomatisation.
Graphs
Here, graphs are undirected and loop-free: G = (V, E) where
E ⊆ V × V is irreflexive and symmetric.
For k ≥ 3, a cycle of length k in G is a sequence v1 , . . . , vk ∈ V of
distinct nodes with (v1 , v2 ), (v2 , v3 ), . . . , (vk , v1 ) ∈ E.
A subset X ⊆ V is independent if E ∩ (X × X) = ∅.
For k < ω, a k-colouring of G is a partition of V into ≤ k independent
sets. The chromatic number χ(G) of G is the least k < ω such that G
has a k-colouring, and ∞ if there is no such k.
Fact 6 A graph has a 2-colouring iff it has no cycles of odd length.
The construction uses graphs of large chromatic number.
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Relation algebras from graphs
Chromatic number and axioms
Given a graph G = (V, E), we can form an atom structure α(G):
,
• The atoms are 1 , rx , bx , gx (for x ∈ V ).
,
• α(G) |= R1, (a) iff a = 1 ,
• α(G) |= R˘(a, b) iff a = b,
• α(G) |= R; (a, b, c) for all atoms a, b, c except where:
,
– one of a, b, c is 1 and the other two are distinct,
– a = rx , b = ry , c = rz , where {x, y, z} is independent,
– similarly for b, g.
(We won’t use these details.)
We write A(G) = Cm α(G) — a kind of ‘Monk algebra’.
Lemma 7 For any graph G, A(G) is a relation algebra.
Theorem 8 (Hirsch–IH) For infinite G, A(G) ∈ RRA iff χ(G) = ∞.
In fact, the higher χ(G) is, the more representable A(G) is:
Proposition 9 There is a special equational axiomatisation
{γn : n < ω} of RRA, obtained by games, such that
1. γn+1 ` γn for all n (the axioms get stronger),
2. For all n < ω there is k < ω such that for all graphs G,
χ(G) ≥ k ⇒ A(G) |= γn ,
3. For all k < ω there is n < ω such that for all infinite G,
A(G) |= γn ⇒ χ(G) ≥ k.
Proof uses Ramsey-type arguments.
The axioms pull the chromatic number along, and vice versa.
What about representability. . . ?
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The example: Erdős graphs
Erdős (1959, probabilistic): for any finite k, there is a finite graph of
chromatic number > k and with no cycles of length < k.
Using a variant of Erdős’s argument, it can be shown that
Proposition 10 For any 2 ≤ k < ω, there is an inverse system
S : G0
π0
←− G1
π1
←− · · ·
of finite graphs Gi with χ(Gi ) = k, and surjective ‘p-morphisms’
πi : Gi+1 →
→ Gi , such that G = lim S is infinite and χ(G) = 2.
←
Duality gives a direct system A(G0 ) → A(G1 ) → · · · of finite relation
algebras and embeddings. Let D = limhA(Gi ) : i < ωi.
→
Lemma 11 (essentially Goldblatt) Dσ ∼
= A(G).
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Theorem 12 (IH–Venema, 2002) There is no canonical first-order
axiomatisation of RRA.
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5. More Erdős graphs!
Worse. . .
These methods have been used in other applications:
Corollary 13 (Venema, 1997) RRA has no Sahlqvist axiomatisation.
[Hirsch–IH, 2002] The class {S : Cm S ∈ RRA} is not elementary.
This answers a question of Maddux (1982).
Can strengthen theorem 12 to show
[Goldblatt–IH–Venema, 2003] There are canonical varieties not
generated by any elementary class of atom structures.
Modal version of this answers a question of Fine (1975).
Theorem 14 RRA has no axiomatisation with only finitely many
non-canonical first-order sentences.
The proof uses an inverse system of finite graphs of chromatic
number k, with inverse limit of chromatic number l (for arbitrary
2 ≤ l ≤ k < ω).
Corollary 15 (Monk, 1964) RRA is not finitely axiomatisable.
Can make a counterexample with relation algebras. Suppose
χ(G) > f (k) ⇒ A(G) |= γk . Let Gk be a finite (Erdős) graph with
χ(G) > f (k) and with no cycles of length < k. Put
K = {A ∈ RA : ∀k < ω(|A| ≥ |A(Gk )| ⇒ A |= γk )}.
Ex: the variety HSPK is canonical but not elementarily generated.
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References
Conclusion
1. P. Erdős, Graph theory and probability, Canad. J. Math. 11
(1959), 34–38.
Monk showed RRA is canonical.
We have seen it’s only barely canonical.
So even this rare silver lining of RRA has its cloud.
But the proof method applies to other interesting results.
Probabilistic constructions may find other applications in algebraic
logic in future.
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2. R. Goldblatt, I. Hodkinson, and Y. Venema, Erdős graphs resolve
Fine’s canonicity problem, Bull. Symbolic Logic 10 (2004),
186–208.
3. R. Hirsch and I. Hodkinson, Strongly representable atom
structures of relation algebras, Proc. Amer. Math. Soc. 130
(2002), 1819–1831.
4. I. Hodkinson and Y. Venema, Canonical varieties with no
canonical axiomatisation, Trans. Amer. Math. Soc. 357 (2005),
4579–4605.
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